**5. Discussions**

**Figure 1.**

 error vs

The physical insights of parameters, used in this research, on velocity, concentration distribution and temperature are key aspects to be discussed in this section. Significance and impact of *γ* (curvature parameter) over the velocity field is shown in Figure 2. It is witnessed from Figure 2 that initially an inverse proportion between *γ* and velocity field as well as boundary layer thickness converts into a direct proportion at far away from the cylinder. The thickness of boundary layer and the velocity distribution with a certain decrease near the cylinder gradually starts increasing in fluid far away from cylinder. More the fluid is near to the cylinder, more is the affect of resistance. Figure 3 shows the behavior of ratio parameter on velocity distribution. Therefore, it can be noticed that velocity field goes higher and higher in both cases for *A* > 1 and *A* < 1 while the boundary layer shows a different behavior. Even At *A* = 1, there are no visuals of boundary layer. The significance of *β*, a third grade parameter, in the fluid velocity is depicted in Figure 4. The more the value of *β*, the low is the viscosity that causes enhancement in velocity distribution. Consequently, the velocity profile is enhanced. Figure 5, depicts the the impact of Reynolds *Re* on the velocity field. Certain decrease in the velocity field is noticed in moving from near the cylinder to away and finally, it vanishes at far away from surface. The reason behind this vanishing is the high value of Reynolds number that reduces the friction in between the surface and fluid. Figure 6 shows the variation of *A*, the ratio parameter, on temperature distribution. Higher is the value of *A*, the lesser is thickness of thermal and temperature boundary layer. Curvature parameter *γ* on *θ* (*η*) is analyzed in Figure 7. Both, the thermal field and connected/associated boundary layer are found as increasing functions of the *γ*. The Impact of well known Prandtl number on *θ* (*η*) is plotted in Figure 8. There is an inverse relation seen in thermal distributions for Prandtl. The smaller is the Prandtl factor, the higher is the temperature and thermal boundary layer thickness. The decremented thermal diffusion due to increment in Prandtl number forces the temperature distribution to decrease. One can conclude that fluids having low Prandtl numbers normally have high thermal diffusivity. The influence of heat generation and absorption on *θ* (*η*) is shown in Figure 9. An increase in heat generation parameter *Q* > 0 and decrease in heat absorption parameter *Q* < 0 ensures the increase in temperature field. Further, the increase in heat generation increases the thickness of thermal boundary layer because the heat generation produces more heat that certainly allows a temperature hike. *Rd* on *θ* (*η*) in Figure 10 shows the variation in temperature distribution due to thermal radiation. More is the thermal radiation *Rd*, lesser is the temperature distribution. The stratification parameter *S* over *θ* (*η*) is analyzed in Figure 11. An increase in *S* certainly decreases the *θ* (*η*) significantly whereas the thickness of thermal boundary layer goes higher and higher over a decrease in *S*. This is justified with the reason that a hike in *S* reduces the difference between surface of cylinder and corresponding temperature. The effect of curvature parameter *γ* over *φ* (*η*) is analyzed in Figure 12. Nearby the cylinder, the concentration profile attains a decrement and goes on increasing away from the cylinder. Figure 13 shows the behavior and variation of Schmidt number *Sc*, ratio of momentum and mass diffusivity, over *φ* (*η*). The more is *Sc*, *φ* (*η*) goes on increasing while thickness of solute boundary layer decreases. Higher is the value of *Sc*, smaller is the mass diffusivity and therefore, *φ* (*η*) achieves an increment. Finally, the variation and behavior of solute stratified parameter *St* over the concentration profile is displayed in Figure 14. Decrements in concentration profile are noted for high values of *St*. Hence, an increase in *St* is responsible for decreasing concentration distribution existent between surface and ambient fluid. Consequently, the concentration field decreases. Optimal convergence control parameters are enlisted in Table 1. It shows the individually calculated average squared r-errors in momentum and energy equations at different order of approximation. A decrease in squared residual errors is noted as compared to the order of approximation. Behavior of the coefficient of skin-friction is enlisted in Table 2. It is evident that for large values of *α*1, *β*, *γ* and *Re*, skin friction increases. The skin friction decreases for augmented values of *α*2 and *A*. Table 3 enlists the variation in Nusselt number due to different parameters. Higher the values of *α*1, *α*2, *β*, *A*, *γ*, *Q*, *St* and *Rd*, higher is the Nusselt number. However, it decreases with Pr. Table 4 shows the influence of numerous parameters on Sherwood number. Higher the values of *S*, *α*1, *α*2, and *β*, higher is the Sherwood number. However, it experiences a decrease in values with *γ*, *St* and *Sc*.

**Figure 2.** Impact of *γ* on *f* (*η*).

**Figure 4.** Impact of *β* on *f* (*η*).

**Figure 6.** Impact of *A* on *<sup>θ</sup>*(*η*).

**Figure 8.** Impact of *Pr* on *<sup>θ</sup>*(*η*).

**Figure 10.** Impact of *Rd* on *<sup>θ</sup>*(*η*).

**Figure 12.** Impact of *γ* on *φ*(*η*).

**Table 1.** Residual errors at *α*1 = *α*2 = *β* = 0.1, *Re*= *γ* = *S* = 0.2, *A* = 1.5, *Rd* = 0.4, *Q* = 0.2, *Sc* = 1.2, *St* = 0.3, Pr = 1 by means of optimal control parameters *f* = −0.32677, *θ* = −0.56129 and *φ* = −0.46129.



**Table 2.** Numerical values of Skin friction for various physical parameters.

**Table 3.** Numerical values of local Nusselt number for various physical parameters.



**Table 4.** Numerical values of local Sherwood number for various physical parameters when *A* = 0.6 and Re = 0.2.
